Abstract

It is known, from results of MacCluer and Shapiro (Canad. J. Math. 38(4):878–906, 1986), that every composition operator which is compact on the Hardy space Hp, 1 ≤ p < ∞, is also compact on the Bergman space \({{\mathfrak B}^p = L^{p}_{a} ({\mathbb D})}\). In this survey, after having described the above known results, we consider Hardy-Orlicz HΨ and Bergman-Orlicz \({{\mathfrak B}^\Psi}\) spaces, characterize the compactness of their composition operators, and show that there exist Orlicz functions for which there are composition operators which are compact on HΨ but not on \({{\mathfrak B}^\Psi}\).

These results come from joint works with P. Lefèvre, H. Queffélec and L. Rodríguez-Piazza ([4–6]). It is an expanded version of the conference I gave at the ICM satellite conference Functional Analysis and Operator Theory, held in Bangalore, India, 8–11 august 2010.